Banach Tarski paradox says that it's possible to decompose a ball in $R^3$ into a finite number of disjoint subsets, which can be then reassembled into 2 identical copies of the original ball. Furthermore, 5 subsets is enough. My questions are:
- How to do this with 5 subsets?
- Is 5 the smallest number?
As I observe, a soccer ball is covered by regular pentagons and regular hexagons (but they are not flat). My questions are:
- How to calculate the number of each type?
- Do these pentagons have the same side? If yes, how long is it? Same questions for hexagons.
- Why are regular pentagon and hexagon but not other regular polygons?, not other combinations?, not other types of polygons?
In both parts, let's use the unit ball (radius = 1) for simplification.